Tensor triangular geometry of band algebras

Abstract

In this paper, we study the tensor triangular geometry of band algebras $RB$, where $B$ is a finite left regular band and $R$ is a commutative Noetherian ring. We show that the Balmer spectrum of $\Perf(RB)$ is homeomorphic to $\Spec R\times L_B$, where $L_B$ is the support lattice of $B$. We also show that there is a bijection between the localizing ideals of $D(RB)$ and the stable subsets of $\Spec R\times L_B$. As a consequence, the telescope conjecture holds for $D(RB)$. Finally, we construct a band $B$ such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category $\Perf(RB)$.